On Geodesic Homotopies of Controlled Width and Conjugacies in Isometry Groups

Groups Geom. Dyn. 7 (2013), 911–929 Groups, Geometry, and Dynamics DOI 10.4171/GGD/210 © European Mathematical Society

On geodesic homotopies of controlled width and conjugacies in isometry groups

Gerasim Kokarev

Abstract. We give an analytical proof of the Poincaré-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two ﬁnite lists in a group acting on an Hadamard space.

Mathematics Subject Classiﬁcation (2010). 53C23, 58E20, 20F65.

Keywords. Homotopy width, harmonic maps, Hadamard space, decision problems.

1. Introduction

Let M and M 0 be smooth Riemannian manifolds without boundary. For a smooth mapping uW M ! M 0 by E.u/ we denote its energy Z E.u/ D kdu.x/k2 dvol.x/; M

0 where the norm of the linear operator du.x/W TxM ! Tu.x/M is induced by the Riemannian metrics on M and M 0. Let u and v be smooth homotopic mappings of 0 M to M and H.s; / be a smooth homotopy between them. The L2-width W2.H / of H is deﬁned as the L2-norm of the function

`H .x/ D the length of the curve s 7! H.s;x/; x 2 M: (1.1)

A smooth homotopy H.s;x/ is called geodesic if for each x 2 M the track curve s 7! H.s;x/ is a geodesic. In [7], [8] Kappeler, Kuksin, and Schroeder prove the following geometric in- 0 equality for the L2-widths of geodesic homotopies when the target manifold M is non-positively curved.

Width Inequality I. Let M and M 0 be compact Riemannian manifolds and suppose that M 0 has non-positive sectional curvature. Let be a homotopy class of maps 912 G. Kokarev

0 of M to M . Then there exist constants C and C with the following property: any smooth homotopic maps u and v 2 can be joined by a geodesic homotopy H whose L2-width W2.H / is controlled by the energies of u and v,

1=2 1=2 W2.H / 6 C.E .u/ C E .v// C C: (1.2)

0 Moreover, if the sectional curvature of M is strictly negative, the constants C and C can be chosen to be independent of a homotopy class .

This inequality can be viewed as a version of the Poincaré inequality for mappings between manifolds. It also has an isoperimetric flavour; it says that the ‘measure’ of the cylinder induced by the homotopy is estimated in terms of the ‘measure’ of its boundary. Inequality (1.2) is a key ingredient in the proof of compactness results for perturbed harmonic map equation [7], [9]. The latter, combined with old results of Uhlenbeck, yields Morse inequalities for harmonic maps with potential [10]. The proof of Width Inequality I in [7], [8] is based on an analogous inequality for maps of metric graphs; see [7], Th. 5.1. In more detail, let G be a finite graph and uW G ! M 0 be a smooth map, that is, whose restriction to every edge is smooth. The length L.u/ of u is defined as the sum of the lengths of the images of the edges. By the L1-width W1.H / of a homotopy H we mean the L1-norm of the length function `H .x/, given by (1.1).

Width Inequality II. Let G be a ﬁnite graph and M 0 be a compact manifold of non-positive sectional curvature. Let be a homotopy class of maps G ! M 0. Then there exist constants C and C with the following property: any smooth homotopic maps u and v 2 can be joined by a geodesic homotopy H such that

W1.H / 6 C.L.u/ C L.v// C C:

0 Moreover, if the sectional curvature of M is strictly negative, the constants C and C can be chosen to be independent of a homotopy class .

The purpose of this note is twofold: firstly, we generalise the width inequalities to the framework of equivariant maps, which we assume to be valued in Hadamard metric spaces. This, in particular, includes width inequalities for homotopies between maps into non-compact target spaces, completing the previous results [8] in this direction. In contrast with the geometric methods in [7], [8] (and also algebraic in [2]), we give an analytical proof of the width inequalities via harmonic map theory. Secondly, we use width inequalities for equivariant maps of trees to study the multiple conjugacy problem for finitely generated groups ƒ acting by isometries on Hadamard spaces. More precisely, under some extra hypotheses, we obtain a linear estimate for the conjugator length in these groups: given two finite conjugate lists of elements .ai /16i6N and .bi /16i6N in ƒ there exists a conjugator g 2 ƒ, On geodesic homotopies of controlled width and conjugacies in isometry groups 913

1 bi D g ai g such that XN jgj 6 C .jai jCjbi j/ C C; iD1 where jj stands for the length d. ;e/in the word metric on ƒ. If the group ƒ has a soluble word problem, then the latter estimate yields immediately the solubility of the multiple conjugacy problem in ƒ. Acknowledgments. The author is grateful to Sergei Kuksin and Dieter Kotschick for their interest in the work and a number of discussions on the subject. The paper was written when the author held a personal EPSRC fellowship at the University of Edinburgh.

2. Statements and discussion of results

2.1. Width inequalities for equivariant maps. Let M be a compact Riemannian manifold without boundary; we denote by Mz its universal cover and by the fundamental group 1.M /. Let .Y; d/ be an Hadamard space; that is a complete length space of non-positive curvature in the sense of Alexandrov (see Section 3 for a precise deﬁnition). Denote by a representation of in the isometry group of Y . Recall that a map uW Mz ! Y is called -equivariant if u.g x/ D .g/ u.x/ for all x 2 M;z g 2 : For -equivariant maps u and v the real-valued functions d.u.x/; v.x//, where x 2 Mz , are invariant with respect to the domain action and, hence, are deﬁned on the quotient M D M=z . In particular, the quantity

Z 1=2 2 d2.u; v/ D d .u.x/; v.x// dvol.x/ (2.1) M defines a metric on the space of locally L2-integrable -equivariant maps. The latter can be also regarded as the L2-width of a unique geodesic homotopy between - equivariant maps. If u is a locally Sobolev W 1;2-smooth -equivariant map, then its energy density measure jduj2 dvol (see Section 3) is also -invariant and the energy of u is defined as the integral Z E.u/ D jduj2 dvol : M Recall that the ideal boundary of Y is defined as the set of equivalence classes of asymptotic geodesic rays; two rays are asymptotic if they remain at a bounded distance from each other. Clearly, any action of by isometries on Y extends to the action on the ideal boundary. 914 G. Kokarev

Theorem 1. Let M be a compact Riemannian manifold without boundary and Y be a locally compact Hadamard space. Let be the fundamental group of M and W ! Isom.Y / be its representation whose image does not ﬁx a point on the ideal boundary of Y . Then there exists a constant C such that for any -equivariant 1;2 locally W -smooth maps u and v the L2-width of a geodesic homotopy H between them satisﬁes the inequality

1=2 1=2 W2.H / 6 C.E .u/ C E .v//: (2.2)

The proof appears in Section 4. The idea is to prove first a similar inequality when one of the maps is an energy minimiser, and then to use compactness properties of the moduli space formed by such minimisers. The former is based on a compactness argument, mimicking the proof of the classical Poincaré inequality. Below we state a version of Theorem 1 for equivariant maps of trees. First, we introduce more notation. Let G be a finite connected graph without terminals and be its fundamental group 1.G/.ByT we denote the universal covering tree of G; the group acts naturally on T by the deck transformations. As above the symbol denotes a representation of in the isometry group of an Hadamard space Y . For a locally rectifiable -equivariant map uW T ! Y , its length density measure jdujdt (see Section 3)is-invariant and the length of u is defined as the integral Z L.u/ D jdujdt: G

Theorem 2. Let G be a finite graph and Y be a locally compact Hadamard space. Let be the fundamental group of G and W ! Isom.Y / be its representation whose image does not fix a point on the ideal boundary of Y . Then there exists a constant C such that for any locally rectifiable -equivariant maps u and v the L1-width of a geodesic homotopy between them satisfies the inequality

W1.H / 6 C.L.u/ C L.v//:

Example. Let M 0 be a non-compact Riemannian manifold whose sectional curvature is negative and bounded away from zero and the injectivity radius is positive. Let 0 W ! 1.M / be a homomorphism whose image is neither trivial nor inﬁnite cyclic. Then the latter does not ﬁx a point on the ideal boundary of the universal cover of M 0. Indeed, the group ./ is generated by hyperbolic elements (regarded as isometries of the universal cover), see [8], Lemma B.1, and the statement follows from the results in [4], Sect. 6. Thus, as a particular case, Theorem 2 contains the width inequality for homotopies between maps from G to M 0; this is the situation considered in [8], Th. 0.1. Under the same hypothesis on M 0, Theorem 1 yields an analogous L2-width inequality for geodesic homotopies between maps from domains whose dimension is greater than one. The latter statement does not seem to be achievable via the methods in [7], [8]. On geodesic homotopies of controlled width and conjugacies in isometry groups 915

We proceed with width inequalities for representations in co-compact subgroups of Isom.Y /. Recall that an action of a group ƒ on a metric space .Y; d/ is said to be co- compact if the quotient Y=ƒis compact. Further, the action of ƒ is said to be proper if for each y 2 Y there exists r>0such that the set fg 2 ƒ j gB.y; r/\B.y; r/ ¤ ¿g is ﬁnite. For a homomorphism W ! ƒ we denote below by Z the centraliser of the image ./ in ƒ.

Theorem 3. Let M be a compact Riemannian manifold without boundary and Y be a locally compact Hadamard space. Let ƒ be a group acting properly and co- compactly by isometries on Y . Denote by the fundamental group of M and let be a homomorphism ! ƒ. Then there are constants C and C such that for any -equivariant locally W 1;2-smooth maps u and v there exists an element h 2 Z such that the L2-width of a geodesic homotopy H between u and h v satisﬁes the inequality 1=2 1=2 W2.H / 6 C.E .u/ C E .v// C C:

Theorem 4. Let G be a finite graph and Y be a locally compact Hadamard space. Let ƒ be a group acting properly and co-compactly by isometries on Y . Denote by the fundamental group of G and let be a homomorphism ! ƒ. Then there are constants C and C such that for any locally rectifiable -equivariant maps u and v there exists an element h 2 Z such that the L1-width of a geodesic homotopy H between u and h v satisfies the inequality

W1.H / 6 C.L.u/ C L.v// C C:

Remark. If the homomorphism W ! ƒ in the theorems is trivial, then the second 1=2 constant C is equal to diam.Y=ƒ/vol M and diam.Y=ƒ/ in the L2- and L1- versions respectively. For non-trivial representations of it can be chosen to be zero.

Example. As a partial case, when the action of ƒ is free, Theorems 3 and 4 above contain width inequalities for homotopies between continuous W 1;2-smooth maps valued in a compact metric space Y=ƒ. The choice of an element h 2 Z in this setting corresponds to the choice of the homotopy between maps. Indeed, recall that the fundamental group of the space formed by continuous maps homotopic to uW M ! Y=ƒ is equal to the centraliser of the image u.1.M // in ƒ.

2.2. Conjugacies of finite lists in isometry groups. Now we describe some ap- plications of the width inequalities to geometric group theory. First, recall that a discrete subgroup ƒ in a Lie group G is called the lattice if the quotient G=ƒ carries a finite G-invariant measure. Such a lattice is always finitely generated provided the group G is semi-simple and has rank > 2; see ref. in [12]. Choose a finite system of generators .gi / of ƒ and consider the word metric d. ; / on ƒ associated with 916 G. Kokarev the Cayley graph determined by the generators. By jgj we denote below the length d.g;e/, the distance between an element g and the neutral element e. The following statements are essentially consequences of Theorems 2 and 4 and are explained in Section 5.

Theorem 5. Let G be a semi-simple Lie group of rank > 2 all of whose simple factors are non-compact. Let ƒ be an irreducible lattice in G and .ai /16i6N be a finite list of elements in ƒ which does not fix a point on the ideal boundary of the associated symmetric space. Then for any conjugate (in ƒ) list .bi /16i6N any 1 conjugating element g 2 ƒ, bi D g ai g satisfies the inequality XN jgj 6 C .jai jCjbi j/ C C; iD1 where the constants depend only on the conjugacy class of the lists. In particular, for such two given lists the set of conjugating elements is finite.

Remark. An analogous statement holds if ƒ is an irreducible lattice in an almost simple p-adic algebraic Lie group of rank > 2. In this case we consider lists which do not ﬁx points on the ideal boundary of the associated Euclidean building.

Example. When the group G is algebraic, the hypothesis on the finite list .ai / is satisfied if, for example, the elements ai ’s generate a lattice (e.g., the whole group ƒ) in G. Indeed, by Borel’s density theorem the latter is Zariski dense in G and, hence, does not fix a point on the ideal boundary of the associated symmetric space.

The estimate above yields immediately an algorithm deciding whether a given list of elements in ƒ is conjugate to the list .ai / in the theorem. This is a special case of the more general result due to Grunewald and Segal [6]: the conjugacy problem for finite lists in arithmetic groups is soluble. (Any irreducible lattice in a semi-simple Lie group of rank > 2 is arithmetic by Margulis’ theorem.) However, we do not know whether the linear estimate for the length of the conjugating element holds under weaker hypotheses than in Theorem 5. We proceed with the conjugacy problem for finite lists in groups which act properly and co-compactly on Hadamard spaces by isometries. Recall that such groups are necessarily finitely presented; see [1], I.8.11. As above by jgj we denote the length d.g;e/ in the word metric.

Theorem 6. Let Y be a locally compact Hadamard space and ƒ be a group acting properly and co-compactly by isometries on Y . Then for any ﬁnite conjugate lists .ai /16i6N and .bi /16i6N of elements in ƒ there exists an element g 2 ƒ with 1 bi D g ai g such that XN jgj 6 C .jai jCjbi j/ C C; (2.3) iD1 On geodesic homotopies of controlled width and conjugacies in isometry groups 917 where the constants depend only on the conjugacy class of the lists. Further, there exists an algorithm deciding whether two given ﬁnite lists of elements in ƒ are conjugate.

When the list .ai / in the theorem consists of a single element, the solubility of the conjugacy problem is well known. It is, for example, a consequence of an exponential bound for the length of the conjugating element in [1], III..1.12. Theorem 6 shows that this exponential bound can be improved to a linear bound (with the cost that the constants become dependant on the conjugacy class), thus, giving a partial affirmative answer to Question 10.4 in [3]. In the context of decision problems it is worth noting that there are finitely presented groups in which the conjugacy problem for elements is soluble, but the conjugacy problem for finite lists is not. We refer to [2] for the explicit examples. Finally, mention that in [2] Bridson and Howie, using purely algebraic methods, prove a closely related linear estimate for the length of the conjugating (two finite lists) element in Gromov hyperbolic groups. An analogous conjugator length estimate is also known for pseudo-Anosov elements in the mapping class group, thus reflecting its hyperbolic character, see [13]. For a more general discussion on the geometry of the conjugacy problem, we refer to the survey talk by Tim Riley [15].

3. Preliminaries

3.1. Sobolev spaces of maps to metric targets. We recall some background mate- rial on Sobolev spaces of maps valued in a metric space. The details can be found in [11]. Let be a Riemannian domain and .Y; d/ be an arbitrary metric space. We suppose that is endowed with a Lebesgue measure dvol induced by the Riemannian volume. A measurable map uW ! Y is called locally L2-integrable if it has a separable essential range and for which d.u. /; Q/ is a locally L2-integrable function on for some Q 2 Y (and, hence, by the triangle inequality for any Q 2 Y ). If the domain is bounded, then the function

Z 1=2 2 d2.u; v/ D d .u.x/; v.x// dvol.x/ deﬁnes a metric on the space of locally L2-integrable maps. The latter is complete provided Y is complete. The approximate energy density of a locally L2-integrable map u is deﬁned for ">0by Z 2 0 d .u.x/; u.x // 0 e".u/.x/ D nC1 dvol.x /; S".x/ " where S".x/ denotes the "-sphere centred at x and n stands for the dimension of . The function e".x/ is non-negative and locally L1-integrable. 918 G. Kokarev

Definition. The energy E.u/ of a locally L2-integrable map u is defined by Z E.u/ D sup lim sup fe".u/ dvol ; 06f 61 "!0 where the sup is taken with respect to compactly supported continuous functions which take values between 0 and 1. A locally L2-integrable map u is called locally 1;2 W -smooth if for any relatively compact domain D the energy E.ujD/ is finite.

Due to the results of Korevaar and Schoen [11], Sect. 1, a locally L2-integrable 1;2 map u is locally W -smooth if and only if there exists a locally L1-integrable function e.u/ such that the measures e".u/ dvol converge weakly to the measure 2 e.u/ dvol as " ! 0. The function e.u/, also denoted by jduR j , is called the energy density of u, and the energy E.u/ is equal to the total mass e.u/ dvol. Now suppose that the domain is 1-dimensional, that is an interval I D .a; b/. For a map uW I ! Y one can also define the approximate length density by d.u.t/; u.t C "// C d.u.t/; u.t "// l".u/.t/ D ;t2 I: " Then the length of u is defined by the formula similar to that for the energy, Z L.u/ D sup lim sup fl".u/dt ; 06f 61 "!0 I where the sup is taken with respect to compactly supported continuous functions. A map uW I ! Y is called rectifiable if its length is finite. In this case there exists aR length density function (or speed function) l.u/ such that the length L.u/ equals l.u/dt.

3.2. Hadamard spaces. Recall that an Hadamard space .Y; d/ is a complete metric space which satisﬁes the following two hypotheses:

(i) Length space. For any two points y0 and y1 2 Y there exists a rectiﬁable curve from y0 to y1 such that

d.y0;y1/ D length./:

We call such a curve geodesic. (ii) Triangle comparison. For any three points P , Q, and R in Y and the choices of geodesics PQ, QR , and RP connecting the respecting points denote by Px, Qx, and Rx the vertices of the (possibly degenerate) Euclidean triangle with the side lengths `.PQ/, `.QR /, and `.RP / respectively. Let Q be a point on the geodesic QR which is a fraction , 0 6 6 1, of the distance from Q to R;

d.Q;Q/D d.Q; R/; d.Q;R/D .1 /d.Q; R/: On geodesic homotopies of controlled width and conjugacies in isometry groups 919

Denote by Qx an analogous point on the side QxRx of the Euclidean triangle. The triangle comparison hypothesis says that the metric distance d.P;Q/ (from Q to the opposite vertex) is bounded above by the Euclidean distance jPx Qxj. This inequality can be written in the form 2 6 2 2 2 dPQ .1 /dPQ C dPR .1 /dQR : (3.1) It is a direct consequence of the property (ii) above that geodesics in an Hadamard space are unique. It is also a consequence of geodesic uniqueness that an Hadamard space has to be simply-connected [1], II.1. Examples include symmetric spaces of non-compact type and Euclidean buildings, simply-connected manifolds of non- positive sectional curvature, Hilbert spaces, simply-connected Euclidean or hyperbolic simplicial complexes satisfying certain local link conditions [1], II.5.4. Another class of examples is provided by the following proposition.

Proposition 1. Let M be a compact Riemannian manifold without boundary and .Y; d/ be an Hadamard space. Let be a representation of the fundamental group D 1.M / in the group of isometries of Y . Then the space of -equivariant locally L2-integrable maps from Mz to Y endowed with the metric (2.1) is an Hadamard space.

The proof follows straightforward from the deﬁnitions: the geodesics in the new space are geodesic homotopies and the triangle comparison hypothesis follows by integration of relation (3.1). A useful consequence of the triangle comparison hypothesis is the following quadrilateral comparison property due to Reshetnyak [14] (we refer to [11], Cor. 2.1.3, for a proof).

Proposition 2. Let .Y; d/ be an Hadamard space and P , Q, R, and S be an ordered sequence of points in Y .For0 6 ; 6 1 deﬁne P to be the point which is the fraction of the way from P to S (on the geodesic PS) and Q to be the point which is the fraction of the way from Q to R (on the opposite geodesic QR ). Then for any 0 6 ˛; t 6 1 the following inequality holds: 2 6 2 2 2 2 dPt Qt .1 t/dPQ C tdRS t.1 t/Œ˛.dPS dQR / C .1 ˛/.dRS dPQ/ :

Setting ˛ to be equal to zero in this inequality, we deduce the convexity of the distance between geodesics

dPt Qt 6 .1 t/dPQ C tdRS : (3.2)

This implies the following energy convexity property. Let u and v be locally W 1;2- smooth maps from the Riemannian domain to an Hadamard space .Y; d/. Let H.s; / be a geodesic homotopy between u and v; the point H.s;x/ is the fraction s 920 G. Kokarev of the way from u.x/ to v.x/, where x 2 . Then for any s the map H.s; / is locally W 1;2-smooth and for any relatively compact domain D its energy satisfies the inequality 1=2 1=2 1=2 E .Hs/ 6 .1 s/E .u/ C sE .v/: (3.3) Inequality (3.2) also yields the length convexity along geodesic homotopies. More precisely, let u and v be rectifiable paths in .Y; d/ and let H.s; / be a geodesic homotopy between the parameterised by the arc length as above. Then for any s the map H.s; / is rectifiable and its length satisfies the inequality

L.Hs/ 6 .1 s/L.u/ C sL.v/: (3.4) Another consequence of the triangle comparison hypothesis is the existence of the nearest point projection W Y ! A onto a convex subset A. In more detail, if .Y; d/ is an Hadamard space and A is its non-empty closed convex subset, then for any y 2 Y there exists a unique point a 2 A which minimises the distance d.y;a/ among all points in A; see [11], Prop. 2.5.4.

3.3. Some properties of harmonic maps. Let M be a compact Riemannian manifold without boundary and .Y; d/ be an Hadamard space. As above by we denote the fundamental group of M and by W ! Isom.Y / its representation in the isometry group of Y . We consider -equivariant locally W 1;2-smooth maps u from the universal cover Mz to Y . The energy density of such a map u is a -invariant function on Mz , which can be also regarded as a function on theR quotient M D M=z .In particular, by the energy E.u/ we understand the integral M e.u/ dvol. We call a - equivariant map harmonic if it minimises the energy among all -equivariant locally W 1;2-smooth maps. The following statement is a straightforward consequence of the energy convexity, formula (3.3). We state it as a proposition for the convenience of references.

Proposition 3. Under the hypotheses above, let u and v be two -equivariant harmonic maps and H.s; / be a geodesic homotopy between them; the point H.s;x/ is the fraction s of the way from u.x/ and v.x/, where x 2 Mz . Then for each s the map H.s; / is also -equivariant harmonic and the energy E.Hs/ does not depend on s.

We proceed with the Lipschitz continuity of harmonic maps. The following proposition is a consequence of the result by Korevaar and Schoen [11], [Th. 2.4.6].

Proposition 4. Under the hypotheses above, any -equivariant harmonic map u is Lipschitz continuous and its Lipschitz constant is bounded above by C E1=2.u/, where the constant C depends on the manifold M and its metric only.

Now let G be a finite connected graph without terminals and be its fundamental group. By T we denote the universal covering tree of G. Similarly to the discussion On geodesic homotopies of controlled width and conjugacies in isometry groups 921 above, for a locally rectifiable -equivariant map uW T ! Y the length density function l.u/ is -invariant and, hence, descends toR the quotient G D T=. In particular, by the length L.u/ we understand the integral G l.u/dt. It is straightforward to see that if a map u minimises the length among all locally rectifiable -equivariant maps, then its restriction to every edge is a geodesic. If the latter has a constant-speed parameterisation on every edge, then it is also harmonic and the length of every edge uI 2 satisfies the relation L .u/ D E.uI /.b a/, see [5], Lemma 12.5. Conversely, if u is a -equivariant harmonic map, then its restriction to every edge is a constant-speed geodesic whose squared length is proportional to the energy as above. In particular, the length is constant on the set of -equivariant harmonic maps, where it achieves its minimum.

4. Proofs of the width inequalities

We start with the following lemma.

Main Lemma I. Let M be a compact Riemannian manifold without boundary and .Y; d/ be a locally compact Hadamard space. Let W ! Isom.Y / be a representation of the fundamental group D 1.M /. Suppose that the moduli space Harm, formed by -equivariant harmonic maps Mz ! Y , is non-empty and bounded in the L2-metric. Then there exists a positive constant C with the following property: for any -equivariant locally W 1;2-smooth map u there exists a harmonic map uN 2 Harm such that 1=2 1=2 d2.u; u/N 6 C.E .u/ E /; (4.1) where E D E.u/N is the energy minimum among -equivariant maps.

Proof. First, note that inequality (4.1) is invariant under the rescaling of the metric on the target space Y . Hence, it is sufﬁcient to prove the lemma under the assumption that the distance d2.u; u/N is not less than one. (4.2)

Suppose the contrary. Then there exists a sequence of maps uk such that

1=2 1=2 d2.uk; u/N > k.E .uk/ E .u//N for any uN 2 Harm. For each uk choose a harmonic map uN k at which the inﬁmum

d2.u; u/N D inffd2.u; v/ j v 2 Harmg is attained. Such a harmonic map uN clearly exists: it is the value of u under the nearest point projection onto Harm. (The lower semicontinuity of the energy [11], Th. 1.6.1, and Proposition 3 imply that Harm is a closed convex subset in the Hadamard space of -equivariant locally L2-integrable maps.) 922 G. Kokarev

k Denote by Hs , where s 2 Œ0; 1, a geodesic homotopy between uN k and uk;we k k set H0 DNuk and H1 D uk. Assuming that the parameter s is proportional to the arc length, we obtain

k k 1=2 1=2 d2.Hs ;H0 / D s d2.uk; uN k/ > s k.E .uk/ E .uN k//:

Recall the energy E1=2. / is convex along geodesic homotopies;

1=2 1=2 1=2 k 1=2 k s.E .uk/ E .uN k// > E .Hs / E .H0 /:

Combining the last two inequalities we conclude that

k k 1=2 k 1=2 k d2.Hs ;H0 / > k.E .Hs / E .H0 //: (4.3)

k k Now choose a sequence of sk 2 Œ0; 1 such that the distance d2.Hsk ;H0 / equals one; by the assumption (4.2) this is possible. Then relation (4.3) implies that the sequence k E.Hsk / converges to E as k !C1. Since the moduli space Harm is bounded in L2-metric, the latter together with the choice of the sk’s implies that the sequence k 1;2 Hsk is bounded in the W -sense; that is k k 6 d2.Hsk ;w/C E.Hsk / C; (4.4) where w isaﬁxed-equivariant map. Now by the version of Rellich’s embedding k theorem [11], Th. 1.13, we can ﬁnd a subsequence Hsk (denoted by the same symbol) 1;2 which converges in L2-metric and point-wise to a locally W -smooth map vN. By the lower semi-continuity of the energy [11], Th. 1.6.1, the map vN is energy minimising and by the point-wise convergence is -equivariant. By the choice of the sk’s we clearly have

k k k k k k d2.H1 ;Hsk / D d2.H1 ;H0 / d2.Hsk ;H0 / D d2.uk; uN k/ 1:

Thus, the L2-distance between the maps uk and v can be estimated by 6 k k k k d2.uk; v/N d2.H1 ;Hsk / C d2.Hsk ; v/N D d2.uk; uN k/ C .d2.Hsk ; v/N 1/: For sufﬁciently large k the second term on the right-hand side is negative, and we arrive at a contradiction with the choice of the harmonic maps uN k’s. The following lemma summarises known results (essentially due to [11]) on the moduli space Harm, formed by -equivariant maps.

Lemma 1. Let M be a compact Riemannian manifold without boundary and Y be a locally compact Hadamard space. Let be the fundamental group of M and W ! Isom.Y / be its representation whose image does not ﬁx a point on the ideal boundary of Y . Then the moduli space Harm, formed by -equivariant harmonic maps, is non-empty and compact in C 0-topology. On geodesic homotopies of controlled width and conjugacies in isometry groups 923

Since there is no direct reference for the statement on the compactness of Harm and to make our paper more self-contained, we give a proof now. Proof of Lemma 1. First, we explain the existence of a -equivariant harmonic map. By [11], Th. 2.6.4, there exists an energy minimising sequence fui g of equivariant Lipschitz continuous maps, whose Lipschitz constants are uniformly bounded. Let be a fundamental domain for the action of on the universal cover Mz . We claim that under the hypotheses of the theorem the ranges ui ./ are contained in a bounded subset of Y . Indeed, suppose the contrary. Then there exists a point x 2 such that the sequence fui .x/g is unbounded in Y , i.e.,

d.ui .x/; Q/ !C1 for some Q 2 Y:

For any g 2 consider the sequence d..g/ ui .x/; ui .x//. By the equivariance of the ui ’s and the uniform boundedness of their Lipschitz constants we have

d..g/ ui .x/; ui .x// 6 Cd.g x;x/; and hence the quantities on the left hand side remain bounded as i !C1. By the convexity of the distance between geodesics, relation (3.2), we see that the (Hausdorff) distances between the geodesic segments Qui .x/ and .g/ Qui .x/ also remain bounded as i !C1. Since Y is locally compact, we can find a subsequence of ui , denoted by the same symbol, such that the segments Qui .x/ converge on compact subsets to a geodesic ray with initial point at Q. Then the distance between and .g/ is also bounded for any g 2 . This shows that represents a fixed point for the action of ./ and leads to a contradiction. Now, since Y is locally compact, the Arzela–Ascoli theorem applies and we can 0 find a subsequence of ui converging in C -topology to an energy-minimising and, hence, harmonic map. Thus, the moduli space Harm is non-empty. Finally, we explain the compactness of Harm. Let ui be a sequence of - equivariant harmonic maps. By Proposition 3 their energies coincide, and by Propo- sition 4 the ui ’s are uniformly Lipschitz continuous. The same argument as above shows that the ranges ui ./ are contained in a bounded subset of Y . Again by the Arzela–Ascoli theorem there exists a converging subsequence. By the lower semicontinuity of the energy the limit map is energy minimising and, hence, harmonic. Thus, the moduli space Harm is compact in C 0-topology among -equivariant harmonic maps. Proof of Theorem 1. By Lemma 1, Main Lemma I applies: for given -equivariant maps u and v we can find harmonic -equivariant maps uN and vN such that d2.u; u/N and d2.v; v/N are estimated as in (4.1). By Lemma 1 the moduli space Harm is compact and, hence, the distance d2.u;N v/N is uniformly bounded. The L2-width of a geodesic homotopy H between u and v is the distance d2.u; v/, and by the triangle inequality we have W2.H / 6 d2.u; u/N C d2.u;N v/N C d2.v; v/:N 924 G. Kokarev

The second term is bounded, and the ﬁrst and the last can be estimated as in (4.1); thus, we obtain 1=2 1=2 W2.H / 6 C.E .u/ C E .v// C C; 1=2 where C equals diam.Harm/2CE . Since, under the hypotheses of the theorem, the energy minimum E is positive, this inequality can be re-written in the form (2.2).

Now we explain the proof of Theorem 2; it follows essentially the same idea. First, we discuss the version of Main Lemma I.Byd1.u; v/ we denote below the maximum of the distance function between maps u and v.

Main Lemma II. Let G be a ﬁnite graph and .Y; d/ be a locally compact Hadamard space. Let W ! Isom.Y / be a representation of the fundamental group D 1.G/. Suppose that the moduli space Harm, formed by -equivariant harmonic maps T ! Y , is non-empty and compact in C 0-topology. Then there exists a positive constant C with the following property: for any continuous rectiﬁable -equivariant map u there exists a harmonic map uN 2 Harm such that

d1.u; u/N 6 C.L.u/ L/; (4.5) where L D L.u/N is the length minimum among -equivariant maps.

Proof. First, without loss of generality we may assume that the maps uW T ! Y under consideration are such that their restrictions to every edge are parameterised proportionally to the arc length. Second, as in the proof of Main Lemma I,itis sufﬁcient to prove the lemma under the assumption that the distance d1.u; u/N is not less than one. Suppose the contrary. Then there exists a sequence of maps uk and harmonic maps uN k such that d1.uk; uN k/ > k.L.uk/ L/I we suppose that the uN k’s minimise the distance fd1.uk;v/, where v 2 Harmg. k Denote by Hs , where s 2 Œ0; 1, a geodesic homotopy between uN k and uk. Assuming that the parameter is proportional to the arc length and using the convexity of the length, relation (3.4), we obtain

k k k d1.Hs / > k.L.Hs / L.H0 //:

Choosing a sequence sk 2 Œ0; 1 such that the left-hand side above equals one, we k k conclude that L.Hsk / converges to L as k !C1. Since the lengths of Hsk are k bounded and the edges of the Hsk ’s are parameterised proportionally to the arc length, k we see that the sequence of the Hsk ’s is equicontinuous. Further, the compactness of Harm implies that the latter sequence is d1-bounded. Now the Arzela–Ascoli On geodesic homotopies of controlled width and conjugacies in isometry groups 925 theorem applies and there exists a subsequence converging in d1-metric to a continuous map vN. The map vN is clearly -equivariant and length-minimising. Moreover, it has a constant-speed parameterisation and, hence, is harmonic. Now one gets a contradiction in the same way as in the proof of Main Lemma I. Proof of Theorem 2. First, Lemma 1 carries over the case of -equivariant maps of trees. In more detail, we need to start with a length minimising sequence which is uniformly Lipschitz continuous. The latter can be constructed by re-parameterising any length minimising sequence proportionally to the arc length on every edge. The rest of the proof (of Lemma 1) carries over without essential changes. Now we simply follow the lines in the proof of Theorem 1 and use Main Lemma II instead of Main Lemma I. We proceed with the proofs of Theorems 3 and 4. First, recall some notation. Let ƒ be a group acting properly and co-compactly by isometries on Y . For a homomorphism W ! ƒ by Z we denote the centraliser of the image ./ in ƒ. The group Z acts naturally on the space of -equivariant maps uW Mz ! Y and, in particular, on the moduli space Harm.

Lemma 2. Under the hypotheses of Theorem 3, the moduli space Harm, formed by -equivariant harmonic maps, is non-empty and the quotient Harm=Z is compact in C 0-topology.

Proof. We start with the existence of a -equivariant harmonic map. By [11], Th. 2.6.4, there exists an energy minimising sequence fui g of equivariant Lipschitz continuous maps, whose Lipschitz constants are uniformly bounded. Let and D be fundamental domains for the actions of on Mz and ƒ on Y respectively. Fix a point x 2 . Then there exists a sequence of elements hi 2 ƒ such that the maps hi ui send x into the closure of D. Since the hi ’s are isometries, the sequence fhi ui g is also energy minimising and uniformly Lipschitz continuous. Moreover, since ƒ acts co-compactly, its fundamental domain D is bounded, and the uniform Lipschitz continuity implies that the ranges hi ui ./ are contained in a bounded subset of Y . By the Arzela–Ascoli theorem there exists a subsequence, also denoted by hi ui , converging to a limit map v. Now we deﬁne a homomorphism ' W ! ƒ such that the limit map v is '- equivariant. For this ﬁx a generator g 2 and consider the points

v.g x/ D lim.hi ui /.g x/ and v.x/ D lim.hi ui /.x/; where x 2 . The triangle inequality implies that

1 .hi .g/hi / v.x/ ! v.g x/ as i !C1:

1 Now, since the action of ƒ is proper, the sequence hi .g/hi contains a constant subsequence; we denote it value by '.g/ 2 ƒ. We use the hi ’s of this subsequence 926 G. Kokarev for the same procedure for another generator in . Repeating the process we deﬁne ' on all generators. It then extends as a homomorphism ' W ! ƒ and the map v is '-equivariant. As a result of this procedure, we also have a sequence hi 2 ƒ such that 1 hi .g/hi D '.g/ for any g 2 :

This identity implies that the hi ’s can be written in the form k hNi , where hNi 2 Z, and the element k 2 ƒ conjugates and '. Now, since the sequence hi ui converges to 1 v, the sequence hNi ui converges to k v. Moreover, the latter is energy minimising and is formed by -equivariant maps. Thus, the limit map k1v is a harmonic - equivariant map and the existence is demonstrated. The compactness of Harm=Z follows by the same argument as above with the substitution of the sequence of harmonic maps for the energy minimising sequence fui g. By Proposition 3 the former sequence is also energy minimising, and by Propo- sition 4 is uniformly Lipschitz continuous; the argument above yields a sequence hNi 2 Z such that hNi ui converges to a -equivariant harmonic map.

Proof of Theorem 3. Let H be a fundamental domain for the action of Z on the moduli space Harm. First, Main Lemma I holds under a weaker hypothesis than the L2-boundedness of Harm. More precisely, it is sufﬁcient to assume that the domain H is bounded in the L2-metric. Indeed, since the group Z acts by isometries, one can suppose that the maps uN k’s (in the proof of Main Lemma I) belong to H. The boundedness of the latter is then used to obtain the W 1;2-boundedness of the sequence k Hsk , relation (4.4). The rest of the proof stays unchanged. Now the combination of Lemma 2 and estimate (4.1) yields the statement in the fashion similar to the proof of Theorem 1.

Proof of Theorem 4. First, Main Lemma II holds under a weaker hypothesis than the compactness of the moduli space Harm. Similarly to the above, it is sufﬁcient to assume that a fundamental domain for the action of Z on Harm is compact. Further, Lemma 2 carries over the case of -equivariant maps of trees; the proof follows essentially the same line of argument. The combination of this version of Lemma 2 with estimate (4.5) yields the statement in the same fashion as above.

5. Finitely generated subgroups in isometry groups

Recall that the action of a group ƒ on a metric space .Y; d/ by isometries deﬁnes an orbit pseudo-metric on ƒ:

dy.g; h/ D d.g y;h y/; where g; h 2 ƒ; and y 2 Y is a ﬁxed reference point. For another point yN 2 Y the pseudo-metrics dy and dyN are coarsely isometric; that is there exists a constant C (D 2d.y; y/N ) such On geodesic homotopies of controlled width and conjugacies in isometry groups 927 that dyN .g; h/ C 6 dy.g; h/ 6 dyN .g; h/ C C:

First, we show that the L1-width inequalities imply an estimate for the conjugating element in the orbit pseudo-metric.

Lemma 3. Let G be a semi-simple Lie group all of whose simple factors are non- compact. Let ƒ be an irreducible lattice in G and .ai /16i6N be a finite list of elements in ƒ which does not fix a point on the ideal boundary of the associated symmetric space. Then for any conjugate (in ƒ) list .bi /16i6N any conjugating element g 2 ƒ, 1 bi D g ai g satisfies the inequality XN dy.g; e/ 6 C .dy.ai ;e/C dy.bi ; e//; iD1 where y 2 Y is a reference point, and the constant depends only on the conjugacy class of the list .ai /.

Proof. Let Y be a symmetric space associated with the Lie group G. Under the hypotheses on G, the natural G-invariant Riemannian metric on Y defines a distance d which makes Y into an Hadamard space. N Consider a bouqet of N copies of a circle; denote by DiD1Z its fundamental group and by T its universal cover. Define a homomorphism W ! ƒ by the rule: the generator of the ith copy of Z maps into ai . For a fixed reference point y 2 Y consider the graph in Y whose vertices are points g y, where g is a word in the alphabet .ai /. The edges are geodesic arcs; two points g1 y and g2 y are 1 joined by an edge if and only if g1 g2 is an element ai or its inverse. Suppose that each edge is parameterised proportionally to the arc length. Such a parameterisation defines a -equivariant map uW T ! Y , whose length L.u/ is given by the sum PN iD1 d.ai y;y/. 1 Analogously, for a conjugate list .bi /16i6N one defines a .g g/-equivariant map v W T ! Y , where g is a conjugating element. Note that the map g v is - PN equivariant and its length L.g v/ coincides with L.v/ D iD1 d.bi y;y/. By the hypotheses of the lemma, Theorem 2 applies and we have

d.g y;y/ 6 W1.H / 6 C.L.u/ C L.v//; where H is a homotopy between u and gv. Now the combination with the expressions for the lengths ﬁnishes the proof. Proof of Theorem 5. The statement is a direct consequence of Lemma 3 and the solution of Kazhdan’s conjecture in [12]. The latter says that the word metric (with respect to some ﬁnite set of generators) on an irreducible lattice ƒ is quasi-isometric to the orbit metric (with respect to the action on the associated symmetric space or Euclidean building) provided G is semi-simple and its rank > 2. 928 G. Kokarev

Lemma 4. Let Y be a locally compact Hadamard space and ƒ be a group acting properly and co-compactly by isometries on Y . Then for any ﬁnite conjugate lists .ai /16i6N and .bi /16i6N of elements in ƒ there exists an element g 2 ƒ with 1 bi D g ai g such that XN dy.g; e/ 6 C .dy.ai ;e/C dy.bi ;e//C C; iD1 where y 2 Y is a reference point, and the constants depend only on the conjugacy class of the list .ai /.

Proof. The proof follows the same line of argument as the proof of Lemma 3 with the use of Theorem 4 instead of Theorem 2. Proof of Theorem 6. By the Švarc–Milnor lemma [1], I.8.19, the word and orbit metrics on ƒ are quasi-isometric. The combination of this with Lemma 4 implies the first statement of the theorem. Further, by [1], III..1.4, the word problem in ƒ is soluble. This yields the algorithm deciding the conjugacy of finite lists in the following fashion. If there exists an element conjugating two given lists, then it belongs to the finite subset of ƒ formed by elements satisfying the bound (2.3). Using the solubility of the word problem, the algorithm checks all elements from this finite set.

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Received January 30, 2010; revised January 22, 2012

G. Kokarev, Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 München, Germany E-mail: [email protected]